We don’t need anything at Planck energies to see a conflict between quantum field theories (QFTs) and General Relativity in the face of non-classical behaviour of matter which generates a strong signal in the stress-energy tensor. The main problem is that that such behaviours are “washed out” by the rest of the stress-energy tensor’s coupling to matter that is “in the classical limit”, and we don’t yet know how to recover classical physics from QFTs.
The interesting-to-theorists quantum gravity stuff is near the centres of neutron stars and black holes, and in the distant past of our universe: all regions we have yet to be figure out how to explore. And all those regions are surrounded by extremely hot and noisy matter: the ultradense nuclear pasta and hot atmosphere and so forth in neutron stars, the accretion regions of black holes, and the “fog” of the surface of last scattering (and the equivalent for neutrinos and possibly equivalent(s) for other non-charged particles or whatever the hell dark matter is). The closest neutron stars and black holes are already a depressingly long way away by RADAR-distance, so there’s no real hope of soon sending a probe close to them to try to gather up and report back observations. So theorists who want to abolish General Relativity’s gravitational singularity, or alternatively quantum things like unitarity, are struggling to extract (even in principle) signal from noisy relics of these extreme-and-hidden things.
However, there is still interesting stuff possible to test near us: if we can develop a laboratory-based object with strongly non-classical behaviour even at sufficient mass to influence lab gravimeters (that is, it will rise above the background stress-energy) then there are all sorts of things we can do to test theories of quantum gravity. We expect that semiclassical gravity will be blown away by such an experiment (there is a member of https://www.npl.washington.edu/eotwash/node/1 here on HN, for instance). An example might be to bring a milligram-or-more coherent mass into a superposition for at least a few milliseconds and see which way an array of lab gravimeters point. That’s already a difficult experiment! (the subsection heading is wildly optimistic, but explains the expected result of such an experiment: https://en.wikipedia.org/wiki/Semiclassical_gravity#Experime... )
I’m not certain the Planck length is anything more than mixing some physical constants into a unit of length, rather than something important to nature. It is interesting to see Planck-scale dimensionful units are frame-invariant — they might not be, which is the premise of Doubly-Special Relativity, for instance. On the other hand, Special Relativity and its deformations might not be the right tool anyway, as a Planck scale object may generate enough spacetime curvature (by virtue of the Planck scale object’s local stress-energy) to blow away the global symmetries of the flat spacetime that form the background for these theories (which are NOT background-independent!). We’re not going to be probing Planck scale objects in the near future, and so there are few limits on speculative theories other than re-re-re-reconfirmations of local Lorentz invariance at lower (stress-)energies in a variety of backgrounds, including that of the Standard Model of Cosmology (the red-shifting of spectra from supernovae, the Lyman-alpha forest, and so on), that of effectively flat spacetime (labs on earth, and in space vehicles around the solar system), and that of significantly curved spacetime (spectra of neutron stars, for instance). And that same information is also evidence favouring one QFT: the Standard Model of Particle Physics (“SM”), which incorporates Lorentz invariance (but not gravitation, so SM is background-dependent, and enjoys the fact that General Relativity predicts the correct background for SM in every sufficiently small region in the universe, even if the background between SM-event-at-A and SM-observer-at-B is such that a region covering A and B is clearly not flat spacetime []).
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[] we can adapt to that by using generally covariant physical theories; unfortunately not all physical theories have been “promoted” into general covariance, and anyway tensors are hard so people will use cut-downs of generally covariant theories adapted to some special background. One can study gauge fixing to figure out how well this works for physical theories, and whether the “comma-goes-to-semicolon” rule of “promoting” theories into general covariance actually ends up with something that works in an arbitrary background. Lots of perfectly-good-here-on-Earth physical theories fall apart in different spacetime backgrounds.